Persistence of multi-dimensional degenerate hyperbolic lower dimensional invariant tori in reversible systems
The authors consider reversible systems with multi-dimensional degenerate hyperbolic lower dimensional tori of prescribed frequencies that persist. They focus on the reversible system given by \(\dot{w} = N(w) + (0, 0, g(v), 0) + f(w)) \) where \(w = (x, y, u, v) \in \mathbb{T}^n \times \mathbb{R}^n \times \mathbb{R}^p \times \mathbb{R}^p\), and \(N = (\omega_0 + My + Dv, 0, Cy + Av, Bu)\) where \(M, D, C, A\), and \(B\) are constant matrices. Further, \(g(v) = (v_1^{2d_1+1}, v_2^{2d_2 + 1}, \dots, v_p^{2d_p+1})\) is a higher order term, \(v = (v_1, v_2, \dots, v_p)\), and \(d_1, d_2, \dots, d_p\) are positive integers. The function \(f = (f_1, f_2, f_3, f_4)\) is a small perturbation. Two additional assumptions are made: the matrix \(M\) is nonsingular and a hyperbolicity assumption holds true, so that all the eigenvalues of the matrix \[\begin{pmatrix} 0 & A \\ B & 0 \end{pmatrix}\] have non-zero real parts. The main theorem states that if \(\omega_0\) satisfies a certain Diophantine condition and \(f\) is real analytic on a small disk about the origin with Fourier series norm that be made arbitrarily small on that disk, then the reversible system admits a lower dimensional invariant torus with prescribed frequency \(\omega_0\). The proof uses KAM theory with parameters as well as some topological degree theory.
scientific article; zbMATH DE number 7637390
scientific article